Symmetric Positive Definite (SPD) matrix learning methods have become popular in many image and video processing tasks, thanks to their ability to learn appropriate statistical representations while respecting the Riemannian geometry of the underlying SPD manifold. In this paper we build a Riemannian network to open up a new direction of SPD matrix non-linear learning in a deep architecture. The built network generalizes the Euclidean network paradigm to non-Euclidean SPD manifolds. In particular, we devise bilinear mapping layers to transform input SPD matrices into more desirable SPD matrices, exploit eigenvalue rectification layers to introduce the non-linearity with a non-linear function on the new SPD matrices, and design eigenvalue logarithm layers to perform Log-Euclidean Riemannian computing on the resulting SPD matrices for regular output layers. For training the deep network, we propose a Riemannian matrix backpropagation by exploiting a variant of stochastic gradient descent on Stiefel manifolds where the network weights reside on. We show through experiments that the proposed SPD network can be simply trained and outperform existing SPD matrix learning and state-of-the-art methods in three typical visual classification tasks.

本文介绍了一种基于Riemannian网络架构的SPD矩阵非线性学习方法，使用双线性映射层、特征值矫正层和特征值对数层，使用基于Stiefel流形的变体随机梯度下降法来训练此深度网络。实验证明，该网络简单易用，并在三个典型的视觉分类任务中优于现有的SPD矩阵学习和现有最先进的方法。

一种用于SPD矩阵学习的黎曼网络